| L(s) = 1 | + 8·4-s − 9-s + 40·16-s − 8·36-s − 14·49-s + 160·64-s + 4·79-s − 8·81-s − 44·109-s − 26·121-s + 127-s + 131-s + 137-s + 139-s − 40·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 112·196-s + ⋯ |
| L(s) = 1 | + 4·4-s − 1/3·9-s + 10·16-s − 4/3·36-s − 2·49-s + 20·64-s + 0.450·79-s − 8/9·81-s − 4.21·109-s − 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.33·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 8·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.874360204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.874360204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 149 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.953543864990201790930227194528, −7.58105496153048037897939138870, −7.14976610812732052661262286955, −7.07389981134382342876969136642, −6.93835373060395594446839931738, −6.55915822949561712266315997933, −6.50112457251198326633862832325, −6.26015882596481085760782353611, −6.11197434253668954085507216899, −5.60464270189659349257384020892, −5.53315514815306802555638861047, −5.28799622511672000328454837670, −5.15716764469403218802270585243, −4.47656069846215656846688628702, −4.15073608586646236136054494772, −3.86273446663877394962146602797, −3.45884935686328489063642666979, −3.06405066808034983850390240680, −3.03395338655639476774053463319, −2.71383075929259734154474270323, −2.45586031035094884912343928929, −1.83786437122672824084415151411, −1.83675786878783096955187525278, −1.46220174315032442660824723054, −0.867863961991654545688222548179,
0.867863961991654545688222548179, 1.46220174315032442660824723054, 1.83675786878783096955187525278, 1.83786437122672824084415151411, 2.45586031035094884912343928929, 2.71383075929259734154474270323, 3.03395338655639476774053463319, 3.06405066808034983850390240680, 3.45884935686328489063642666979, 3.86273446663877394962146602797, 4.15073608586646236136054494772, 4.47656069846215656846688628702, 5.15716764469403218802270585243, 5.28799622511672000328454837670, 5.53315514815306802555638861047, 5.60464270189659349257384020892, 6.11197434253668954085507216899, 6.26015882596481085760782353611, 6.50112457251198326633862832325, 6.55915822949561712266315997933, 6.93835373060395594446839931738, 7.07389981134382342876969136642, 7.14976610812732052661262286955, 7.58105496153048037897939138870, 7.953543864990201790930227194528
